Introduction to Graphene-Based Nanomaterials

Transcription

1 Introduction to Graphene-Based Nanomaterials Beginning with an introduction to carbon-based nanomaterials, their electronic properties, and general concepts in quantum transport, this detailed primer describes the most effective theoretical and computational methods and tools for simulating the electronic structure and transport properties of graphene-based systems. Transport concepts are clearly presented through simple models, enabling comparison with analytical treatments, and multiscale quantum transport methodologies are introduced and developed in a straightforward way, demonstrating a range of methods for tackling the modeling of defects and impurities in more complex graphene-based materials. The authors also discuss the practical applications of this revolutionary nanomaterial, contemporary challenges in theory and simulation, and long-term perspectives. Containing numerous problems for solution, real-life examples of current research, and accompanied online by further exercises, solutions, and computational codes, this is the perfect introductory resource for graduate students and researchers in nanoscience and nanotechnology, condensed matter physics, materials science, and nanoelectronics. Luis E. F. Foa Torres is a Researcher at the Argentine National Council for Science and Technology (CONICET) and an Adjoint Professor at the National University of Córdoba, Argentina, specializing in quantum transport with emphasis on inelastic effects anddrivensystems. Stephan Roche is an ICREA Research Professor at the Catalan Institute of Nanoscience and Nanotechnology (ICN2), where he is Head of the Theoretical and Computational Nanoscience Group, focusing on quantum transport phenomena in materials such as graphene. Jean-Christophe Charlier is a Professor of Physics at the University of Louvain, Belgium, whose interests include condensed matter physics and nanosciences. His main scientific expertise focuses on first-principles computer modeling for investigating carbon-based nanomaterials.

2 Torres, Roche, and Charlier have written a very attractive book on graphene-based materials that takes a reader or student with no prior exposure to this topic to a level where he or she can carry out research at a high level and work in this area professionally, assuming a standard background of a condensed matter physics graduate student. The material is nicely organized into chapters which can be subdivided into daily learning segments and with problem sets that could be helpful for either formal course presentation or self study. Four appendices with more detailed presentations allow readers to develop the skills needed for using and extending present knowledge to advancing the science of few-layered materials or for developing applications based on these materials. All in all I would expect this to become a popular text for present and future researchers who will be active in the present decade, advancing science and launching technological innovation. Mildred Dresselhaus, Massachusetts Institute of Technology This book covers the fundamental aspects of graphene, starting from the very beginning. By reading this book, most basic subjects on graphene and some special theoretical methods can be understood at a high level. Starting with the current status of graphene science, the authors proceed to a self-contained description of the electronic structure of graphene, then an especially detailed description of the transport properties of graphene, such as back scattering, Klein tunnelling, quantum dots, Landau levels etc., based on the authors work. Methods introduced to investigate these subjects range from the tight binding method to ab initio calculations, so that the readers can select their preferred method, and the appendices contain useful mathematical explanations for these methods. Thus, without reading the other textbooks, the reader can understand the text. The book should be useful not only for theoretical researchers but also for graduate students and experimental researchers, who will quickly understand the theorists perspective. This book will be an important basic textbook on the physics of graphene. R. Saito, Tohoku University

3 Introduction to Graphene-Based Nanomaterials From Electronic Structure LUIS E. F. FOA TORRES Argentine National Council for Science and Technology (CONICET) and National University of Córdoba, Argentina STEPHAN ROCHE ICREA and Catalan Institute of Nanoscience and Nanotechnology JEAN-CHRISTOPHE CHARLIER University of Louvain, Belgium

4 University Printing House, Cambridge CB2 8BS, United Kingdom Published in the United States of America by Cambridge University Press, New York Cambridge University Press is part of the University of Cambridge. It furthers the University s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. Information on this title: / Foa Torres, Roche & Charlier 2014 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2014 Printed in the United Kingdom by TJ International Ltd. Padstow Cornwall A catalog record for this publication is available from the British Library Library of Congress Cataloging-in-Publication Data Foa Torres, Luis E. F. Introduction to graphene-based nanomaterials: from electronic structure to quantum transport / Luis E. F. Foa Torres, Stephan Roche, Jean-Christophe Charlier. pages cm ISBN (hardback) 1. Nanostructured materials. 2. Graphene. 3. Quantum theory. I. Roche, Stephan. II. Charlier, Jean-Christophe. III. Title. TA418.9.N35F dc Additional resources for this publication at /foatorres Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

5 Contents Preface page xi 1 Introduction to carbon-based nanostructures Carbon structures and hybridizations Carbon nanostructures Guide to the book Further reading 10 2 Electronic properties of carbon-based nanostructures Introduction Electronic properties of graphene Tight-binding description of graphene Effective description close to the Dirac point and massless Dirac fermions Electronic properties of graphene beyond the linear approximation Electronic properties of few-layer graphene Electronic properties of graphene nanoribbons Electronic properties of armchair nanoribbons (agnrs) Electronic properties of zigzag nanoribbons (zgnrs) Electronic properties of carbon nanotubes Structural parameters of CNTs Electronic structure of CNTs within the zone-folding approximation Curvature effects: beyond the zone-folding model Small-diameter nanotubes: beyond the tight-binding approach Nanotubes in bundles Multiwall nanotubes Spin orbit coupling in graphene Magnetic field effects in low-dimensional graphene-related materials Short historical perspective Peierls substitution 58

6 vi Contents Parallel field, Aharonov Bohm gap opening and orbital degeneracy splitting Perpendicular field and Landau levels Landau levels in graphene Defects and disorder in graphene-based nanostructures Structural point defects in graphene Grain boundaries and extended defects in graphene Structural defects at graphene edges Defects in carbon nanotubes Further reading and problems 85 3 Quantum transport: general concepts Introduction Relevant time and length scales Coherent versus sequential transport Landauer Büttiker theory Heuristic derivation of Landauer s formula Boltzmann semiclassical transport The relaxation time approximation and the Boltzmann conductivity Kubo formula for the electronic conductivity Illustrations for ballistic and diffusive regimes Kubo versus Landauer Validity limit of Ohm s law in the quantum regime The Kubo formalism in real space Scaling theory of localization Quantum transport beyond the fully coherent or decoherent limits Further reading and problems Klein tunneling and ballistic transport in graphene and related materials The Klein tunneling mechanism Klein tunneling through monolayer graphene with a single (impurity) potential barrier Klein tunneling through bilayer graphene with a single (impurity) potential barrier Ballistic transport in carbon nanotubes and graphene Ballistic motion and conductance quantization Mode decomposition in real space Fabry-Pérot conductance oscillations Contact effects: SWNT-based heterojunctions and the role of contacts between metals and carbon-based devices 135

7 Contents vii 4.3 Ballistic motion through a graphene constriction: the 2D limit and the minimum conductivity Further reading and problems Quantum transport in disordered graphene-based materials Elastic mean free path Temperature dependence of the mean free path Inelastic mean free path in the high-bias regime Quantum interference effects and localization phenomena in disordered graphene-based materials Edge disorder and transport gaps in graphene nanoribbons Transport properties in disordered two-dimensional graphene Two-dimensional disordered graphene: experimental and theoretical overview Metallic versus insulating state and minimum conductivity Boltzmann transport in two-dimensional graphene Kubo transport: graphene with Anderson disorder Kubo transport: graphene with Gaussian impurities Weak localization phenonema in disordered graphene Strong localization in disordered graphene Quantum Hall effect in graphene Hall quantization in graphene The mystery of the zero-energy Landau level splitting Universal longitudinal conductivity at the Dirac point Graphene with monovacancies Electronic structure of graphene with monovacancies Transport features of graphene with monovacancies Polycrystalline graphene Motivation and structural models Electronic properties of polycrystalline graphene Mean free path, conductivity and charge mobility Amorphous graphene Structural models Electronic properties of amorphous graphene Mean free path, conductivity and localization Phonon transport in graphene-related materials Computational phonon propagation methodology Disordered carbon nanotubes with isotope impurities Disordered graphene nanoribbons with edge disorder Graphene quantum dots Generalities on Coulomb blockade Confining charges in graphene devices Further reading and problems 217

8 viii Contents 6 Quantum transport beyond DC Introduction: why AC fields? Adiabatic approximation Floquet theory Average current and density of states Homogeneous driving and the Tien Gordon model Time-evolution operator Overview of AC transport in carbon-based devices AC transport and laser-induced effects on the electronic properties of graphene Further reading and problems Ab initio and multiscale quantum transport in graphene-based materials Introduction Chemically doped nanotubes Tight-binding Hamiltonian of the pristine carbon nanotube Boron-doped metallic carbon nanotubes Nitrogen-doped metallic carbon nanotubes Two-dimensional disordered graphene with adatoms defects Monatomic oxygen defects Atomic hydrogen defects Scattering times Structural point defects embedded in graphene Ab initio quantum transport in 1D carbon nanostructures Introduction Carbon nanotubes Defective carbon nanotubes Doped carbon nanotubes Functionalized carbon nanotubes Carbon nanotubes decorated with metal clusters Graphene nanoribbons Graphene nanoribbons with point defects Graphene nanoribbons with edge reconstruction Graphene nanoribbons with edge disorder Doped graphene nanoribbons GNR-based networks Conclusion Applications Introduction Flexible electronics High-frequency electronics 302

9 Contents ix 8.4 Optoelectronics photonics plasmonics Opacity of graphene and fine structure constant Harnessing graphene with light: a new dimension of possibilities Digital logic gates Digital nonvolatile graphene memories Graphene nanoresonators Spintronics Further reading 313 Appendix A Electronic structure calculations: the density functional theory (DFT) 314 A.1 Introduction 314 A.2 Overview of the approximations 314 A.2.1 The Schrödinger equation 314 A.2.2 The Born Oppenheimer approximation 315 A.2.3 The Hartree approximation 316 A.2.4 The Hartree Fock approximation 317 A.3 Density functional theory 318 A.3.1 The Thomas Fermi model 318 A.3.2 The Hohenberg Kohn theorem 319 A.3.3 The Kohn Sham equations 320 A.3.4 The exchange correlation functionals 322 A.4 Practical calculations 324 A.4.1 Crystal lattice and reciprocal space 324 A.4.2 The plane wave representation 325 A.4.3 k-point grids and band structures 326 A.4.4 The pseudopotential approximation 326 A.4.5 Available DFT codes 330 Appendix B Electronic structure calculations: the many-body perturbation theory (MBPT) 332 B.1 Introduction 332 B.2 Many-body perturbation theory (MBPT) 333 B.2.1 Hedin s equations 333 B.2.2 GW approximation 334 B.3 Practical implementation of G 0 W B.3.1 Perturbative approach 335 B.3.2 Plasmon pole 336 Appendix C Green s functions and ab initio quantum transport in the Landauer Büttiker formalism 338 C.1 Phase-coherent quantum transport and the Green s function formalism 338 C.2 Self-energy corrections and recursive Green s functions techniques 344

10 x Contents C.3 Dyson s equation and an application to treatment of disordered systems 347 C.4 Computing transport properties within ab initio simulations 351 Appendix D Recursion methods for computing the DOS and wavepacket dynamics 358 D.1 Lanczos method for the density of states 358 D.1.1 Termination of the continued fraction 361 D.2 Wavepacket propagation method 363 D.3 Lanczos method for computing off-diagonal Green s functions 367 References 370 Index 405

11 Preface Once deemed impossible to exist in nature, graphene, the first truly two-dimensional nanomaterial ever discovered, has rocketed to stardom since being first isolated in 2004 by Nobel Laureates Konstantin Novoselov and Andre K. Geim of the University of Manchester. Graphene is a single layer of carbon atoms arranged in a flat honeycomb lattice. Researchers in high energy physics, condensed matter physics, chemistry, biology, and engineering, together with funding agencies, and companies from diverse industrial sectors, have all been captivated by graphene and related carbon-based materials such as carbon nanotubes and graphene nanoribbons, owing to their fascinating physical properties, potential applications and market perspectives. But what makes graphene so interesting? Basically, graphene has redefined the limits of what a material can do: it boasts record thermal conductivity and the highest current density at room temperature ever measured (a million times that of copper!); it is the strongest material known (a hundred times stronger than steel!) yet is highly mechanically flexible; it is the least permeable material known (not even helium atoms can pass through it!); the best transparent conductive film; the thinnest material known; and the list goes on... A sheet of graphene can be quickly obtained by exfoliating graphite (the material that the tip of your pencil is made of) using sticky tape. Graphene can readily be observed and characterized using standard laboratory methods, and can be mass-produced either by chemical vapor deposition (CVD) or by epitaxy on silicon carbide substrates. Driven by these intriguing properties, graphene research is blossoming at an unprecedented pace and marks the point of convergence of many fields. However, given this rapid development, there is a scarcity of tutorial material to explain the basics of graphene while describing the state-of-the-art in the field. Such materials are needed to consolidate the graphene research community and foster further progress. The dearth of up-to-date textbooks on the electronic and transport properties of graphene is especially dramatic: the last major work of reference in this area, written by Riichiro Saito, Gene Dresselhaus, and Mildred Dresselhaus, was published in Seeking to answer the prayers of many colleagues, who have had to struggle in a nascent field characterized by a huge body of research papers but very little introductory material, we decided to write this book. It is the fruit of our collective research experience, dating from the early days of research on graphene and related materials, up through the past decade, when each of us developed different computational tools and

12 xii Preface theoretical approaches to understand the complex electronic and transport properties in realistic models of these materials. We have written Introduction to Graphene-Based Nanomaterials: From Electronic Structure for everyone doing (or wishing to do) research on the electronic structure and transport properties of graphene-related systems. Assuming basic knowledge of solid state physics, this book offers a detailed introduction to some of the most useful methods for simulating these properties. Furthermore, we have made additional resources (computational codes, a forum, etc.) available to our readers at cambridge.org/foatorres, and at the book website (introductiontographene.org), where additional exercises as well as corrections to the book text (which will surely appear) will be posted. Graphene and related materials pertain to a larger family that encompasses all kinds of two-dimensional materials, from boron nitride lattices, to transition-metal dichalcogenides (MoS 2,WS 2 ), to the silicon analogue of graphene, silicene, a recently discovered zero-gap semiconductor. Researchers are beginning to explore the third dimension by shuffling two-dimensional materials and by fabricating three-dimensional heterostructures(bn/graphene,bn/mos 2 /graphene,etc.)withunprecedentedproperties. Interestingly, low-energy excitations in two-dimensional graphene (and in onedimensional metallic carbon nanotubes), known as massless Dirac fermions, also develop at the surface of topological insulators (such as BiSe 2,Bi 2 Te 3, etc.), which are bulk insulators. Topological insulators thus share commonalities with graphene, such as Berry s phase-driven quantum phenomena (Klein tunneling, weak antilocalization,...), and exhibit other features such as spin-momentum locking that offer different and ground-breaking perspectives for spintronics. Therefore, we believe that our presentation of the fundamentals of electronic and transport properties in graphene and related materials should prove useful to a growing community of scientists, as they touch on advanced concepts in condensed matter physics, materials science, and nanoscience and nanotechnology. The book starts with an introduction to the electronic structures and basic concepts in transport in low-dimensional materials, and then proceeds to describe the specific transport phenomena unique to graphene-related materials. Transport concepts are then presented through simple disorder models, which in some cases enable comparison with analytical treatments. Additionally, the development of multiscale quantum transport methodologies (either within the Landauer Büttiker or Kubo Greenwood formalisms) is introduced in a straightforward way, showing the various options for tackling defects and impurities in graphene materials with more structural and chemical complexity: from combined ab initio with tight-binding models, to transport calculations fully based on first principles. To facilitate reading, the essential technical aspects concerning the formalism of Green functions, as well as transport implementation and order-n transport schemes are described in dedicated appendices. This book encompasses years of scientific research that has enabled us to establish certain foundations in the field, a work made possible by the efforts of collaborators, including many postdoctoral and doctoral students. We are particularly indebted to Hakim Amara, Rémi Avriller, Blanca Biel, Andrés Botello-Méndez, Victoria Bracamonte, Hernán Calvo,

13 Preface xiii Jessica Campos-Delgado, Damien Connétable, Alessandro Cresti, Eduardo Cruz-Silva, Aron Cummings, Virginia Dal Lago, Xavier Declerck, Simon Dubois, Nicolas-Guillermo Ferrer Sanchez, Lucas Ingaramo, Gabriela Lacconi, Sylvain Latil, Nicolas Leconte, Aurélien Lherbier, Alejandro Lopez-Bezanilla, Thibaud Louvet, Yann-Michel Niquet, Daijiro Nozaki, César Núñez, Hanako Okuno, Frank Ortmann, Andreï Palnichenko, Pablo Pérez-Piskunow, Juan Pablo Ramos, Claudia Rocha, Xavier Rocquefelte, Luis Rosales, Haldun Sevincli, Eric Suárez Morell, Florent Tournus, François Triozon, Silvia Urreta, Gregory Van Lier, Dinh Van Tuan, François Varchon, Wu Li, Zeila Zanolli, and Bing Zheng. We would also like to express our sincere gratitude to the following inspiring individuals with whom we have worked over the past decade: Pulickel Ajayan, Tsuneya Ando, Marcelo Apel, Adrian Bachtold, Carlos Balseiro, Florian Banhart, Robert Baptist, Christophe Bichara, Xavier Blase, Roberto Car, Antonio Castro-Neto, Mairbek Chshiev, Gianaurelio Cuniberti, Silvano De Franceschi, Hongjie Dai, Alessandro De Vita, Millie and Gene Dresselhaus, François Ducastelle, Reinhold Egger, Peter Eklund, Morinobu Endo, Walter Escoffier, Chris Ewels, Andrea Ferrari, Albert Fert, Takeo Fujiwara, Xavier Gonze, Andrea Latgé, Caio Lewenkopf, Annick Loiseau, Jose-Maria Gómez Rodriguez, Nicole Grobert, Paco Guinea, Luc Henrard, Eduardo Hernández, Jean-Paul Issi, Ado Jorio, Philip Kim, Jani Kotakoski, Vladimir Kravtsov, Philippe Lambin, Sergio Makler, Ernesto Medina, Vincent Meunier, Natalio Mingo, Costas Moulopoulos, Joel Moser, Yann-Michel Niquet, Kentaro Nomura, Kostya Novoselov, Pablo Ordejón, Pedro Orellana, Mónica Pacheco, Horacio Pastawski, Marcos Pimenta, Bertrand Raquet, Gian-Marco Rignanese, Angel Rubio, Riichiro Saito, Bobby Sumpter, Mauricio and Humberto Terrones, Gonzalo Usaj, and Sergio Valenzuela. We thank our home institutions for supporting our research, as well as the Alexander von Humboldt Foundation (SR and LEFFT) and the Abdus Salam International Centre for Theoretical Physics (LEFFT). Finally, we are indebted to our respective wives (Sandra Rieger, Encarni Carrasco Perea and Mireille Toth-Budai) and our children (Hector and Gabriel Roche, and Ilona, Elise, and Mathilde Charlier) for their warm enthusiasm and continuous support during all these years of time-consuming work. We hope that you find this book to be a useful companion for starting in this field and perhaps even for your day-to-day research. We recommend that you start by reading Chapter 1 and then follow the advice in the Guide to the book (Section 1.3). And we wish you an exciting journey in Flatland!... Luis E. F. Foà Torres, Stephan Roche, and Jean-Christophe Charlier.